Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $r \neq 0$. $y = \dfrac{20r - 30}{10} \div \dfrac{4r(2r - 3)}{10} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{20r - 30}{10} \times \dfrac{10}{4r(2r - 3)} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (20r - 30) \times 10 } { 10 \times 4r(2r - 3) } $ $ y = \dfrac {10 \times 10(2r - 3)} {10 \times 4r(2r - 3)} $ $ y = \dfrac{100(2r - 3)}{40r(2r - 3)} $ We can cancel the $2r - 3$ so long as $2r - 3 \neq 0$ Therefore $r \neq \dfrac{3}{2}$ $y = \dfrac{100 \cancel{(2r - 3})}{40r \cancel{(2r - 3)}} = \dfrac{100}{40r} = \dfrac{5}{2r} $